Find all exact angles x in the interval [0,360°] that satisfy the following equations without the use of calculator a)sin x +cos 2x=0 b)cos 2x= sin52 ?

1 Answer
Nghi N.
Aug 30, 2015

Solve sin x + cos 2x = 0

Ans: #pi/2; (7pi)/6; (11pi)/6#

Explanation:

Replace in the equation #cos 2x# by #(1 - 2sin^2 x)#, then by changing side, we get a quadratic equation;
#2sin^2 x - sin x - 1 = 0#
Since (a + b + c = 0), use the shortcut. The 2 real roots are: #sin x = 1 # and #sin x = c/a = - 1/2#
a. sin x = 1 --> #x = pi/2#
b. #sin x = -1/2# --> #x = -pi/6 (or (11pi)/6)# and #x = (7pi)/6#
Ans: #pi/2; (7pi)/6, (11pi)/6#
Check by calculator.
x = 90 -> sin x = 1, cos 2x = cos 180 = -1 --> 1 - 1 = 0. OK
x = (7pi/6) = 210 -> sin x = -0.5; cos 2x = cos 420 = cos 60 = 0.5 ->
Therefor, -0.5 + 0.5 = 0. OK
x = (11pi)/6 -> sin 330 = -0.5; cos 660 = cos 300 = 0.5.
Therefor, -0.5 + 0.5 = 0. OK

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